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OK. Welcome ladies and gentlemen. So what I'd like to do is show you how to determine the focus, vertices, as well as co-vertices of an ellipse when the center is at the origin. And again, we know that the center is at the origin because the center, labeled up here, is h, k. And for the formulas here, I have x minus h, y minus k. And none of these have any as a fraction. So therefore, the center is at the origin.
I also wrote up the general equation for a ellipse when it has a horizontal major axis. I wrote the equation of an ellipse for a vertical major axis as well as I, again, reiterated what a represented, what b represents, what c represents as well as the relationship between a, b, and c.
OK, so when doing these problems, a couple things that I always look for. Remember that a is the endpoint of our major axis. And the major axis is always larger than the minor axis. It doesn't matter if it's vertical or horizontal. I'm just going to move my hands. But the major axis is always larger than the minor axis. That means a is always larger than b. So therefore, a squared has to be always larger than b squared.
So when I look at a problem, the first thing I want to do is identify a squared and b squared. Well, in this example, my a squared and my b squared's are my denominators, right? It's either a squared, b squared, or b squared, a squared. Since 25 is larger than 16, I can determine-- let's use a different color-- that a squared is equal to 25. And b squared is equal to 16.
Now, to find my vertices, my co-vertices, and my center, I need to figure out what a, b, and c are. So a squared and b squared aren't going to help me. I need to figure out what a and b are. Well, I don't really need to do a lot of math. I can just say if a squared is 25, then I know a is 5. And if b squared is 16, then I know b is 4.
Now typically, at this point in time, I always like to say, well, I might as well figure out what c is. So I go to my relationship. And I have c squared equals a squared minus b squared. And I just plug in my values for a squared and b squared.
So, therefore, I have c squared equals 25 minus 16. C squared equals 9. So therefore, c is equal to 3. So pretty cool and pretty quick. I was able to identify a, b, and c. That doesn't happen very often that easily.
The next thing, to identify the vertices, co-vertices, and the center, I like to draw a-- I like to draw a picture or at least a graph. I like to plot the information. All right. So I'm just going to draw a nice coordinate axis. I know that the center is at 0, 0 because that information was given to me.
Now, remember, a squared was 25. Since a squared is under the x, that means my major axis is horizontal. If a squared is under the y, that means your major axis is vertical. So a lot of times, what I think is helpful, is sometimes just drawing like a nice little dashed line to remember that that is the major axis.
Now, you don't need to write the minor axis, but just so you know, the minor axis is perpendicular. And where the minor and the major axes intersect is the center. All right. But it's important.
A lot of times, why I like writing that, is because what lies on the major axis? Well, the vertices are the endpoints of the major axis. The center lies on the major axis as well as the foci. The minor axis is going to be your two co-vertices are the endpoints of your minor axis.
All right. So we know that a is equal to 5. A represents the distance to the vertices from the center. So here's my center. Well, one vertices is going to be to the right, 1, 2, 3, 4, 5. Label that vertice.
The other vertice is going to be to the left, 1, 2, 3, 4, 5. Because a represents a distance in the positive and in the negative direction, but it's a length, so it's never going to be negative. It's just in the negative directions. It goes to the right and to the left.
Again, think about like a picture. Here's an ellipse. There's your major axis. You have two endpoints. Here's your center. You're going to the right. You're going to the left.
OK, so now I can also-- well, I'll get to that in a second. The next one is let's go ahead and find our foci. Foci is 3. 3 is a distance to the foci from the center. Remember though, the foci also lie on the major axis. So that's 1, 2, 3. 1, 2, 3.
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